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Transcript
Math-Handout Lars Forsberg Uppsala University, Department of Statistics October 28, 2012 Abstract This note is a brief summary of the introduction to math/calculus presented as a part of B1 - Probability Theory and Statistical Inference, Fall 2012. 0.1 Quadratic identities For and being real numbers, then ( + )2 = 2 + 2 + 2 ( − )2 = 2 + 2 − 2 ( + ) ( − ) = 2 − 2 and ( + + )2 = 2 + 2 + 2 + 2 + 2 + 2 0.2 Power function The general power function in given by () = 1 where and are constants. Some rules for manipulating power functions are = + () = µ ¶ = ( ) = = − −1 = 1 − = 1 √ = 12 1 √ = −12 0.3 Exponential function For the exponential function () = 2 note that is a constant and that is the variable. We have that = + ( ) = () = = − ³ ´ = 0.4 The logarithmic function Definition of ln is given by ln = 0 Now, it follows that ln 1 = 0 ln = 1 Some rules for logarithmic function ln ln () ln ln ln = ln + ln = ln − ln = ln = 3 1 Differentiation In the following, let () and () both be differentiable at the point Here, some rules for differentiation () = ⇒ 0 () = −1 () = 1 ⇒ 0 () = − 12 () = √ () = ⇒ 0 () = ⇒ 0 () = ln ( 0) () = ⇒ 0 () = () = ln 1.1 1 √ 2 ( 0) ⇒ 0 () = 1 Power rule For being constant, then () = ⇒ 0 () = −1 1.2 The derivative of Sums and differences If () = () ± () then 0 () = 0 () ± 0 () 1.3 The derivative of a product If () = () () then 0 () = 0 () () + () 0 () 4 1.4 The derivative of a quotient If () 6= 0 and () = then 0 () = 1.5 () () 0 () () − () 0 () ( ())2 The Chain rule Let = ( ()) then 0 = 0 ( ()) 0 () where 0 () is the ‘inner derivative’. In Leibnizs’ notation, = () and = (), then = 2 Using derivatives to find extreme values of functions One important application of derivatives is to find minium and maximum of functions. Below a brief review of these concepts. In statistics we apply these techniques when estimating parameters of distributions. In Ordinary Least Squares (OLS) we want to find minimum, and in Maximum Likelihood (ML) we want to find maximum. 2.1 Finding minimum To find a minimum of a function () then do the following: 1. Take derivative of () i.e. find 0 () 5 2. Set 0 () = 0 3. Solve for , gives 0 4. Check second derivative of the function at 0 now if 00 () 0 then = 0 is a (local) minimum. 2.2 Finding maximum To find a maximum of a function () then do the following 1. Take derivative of () i.e. find 0 () 2. Set 0 () = 0 3. Solve for , gives 0 4. Check second derivative of the function at 0 now if 00 () 0 then = 0 is a (local) maximum. 6 3 Integration Let () be a function of then the indefinite integral of () is given by Z () = () + where is any constant. 3.1 Some Primitive functions Let () a function and then denote its primitive function by () then ⇒ () = + () = () = () = 1 ( 6= −1) ⇒ () = ( 6= 0) () = 3.2 + ⇒ () = ln + () = () = +1 +1 ⇒ () = + ( 0 6= 1) ⇒ () = 1 (−) ln + ⇒ () = ln | − | + Integration by parts The formula for integration by parts Z Z 0 () () = () () − 0 () () 3.3 Integration by substitution Integration by substitution, or by change of variable is done as follows Z Z 0 [ ()] () = () where = () 7 3.4 Some general rules The area under a curve, (), is calculated with the aid of integrals. Z The area under () for ≤ ≤ = () = () − () where () is the primitive function of (). For constants and we have Z () = Z () Integral of a sum is the sum of the integrals, that is Z [ () + ()] = Z () + Z () For any constant Z () = Z () + Z () also, for ≤ we have Z Z () = − () 8
